7. Melde\'s exp - manual of melde\'s experiment PDF

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manual of melde's experiment...

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Experiment-7 Melde’s experiment

Objective:

To determine the frequency of AC mains by Melde€s experiment.

Apparatus: weights 1gm, 2gm, 5gm, AC Power supply 6V, 500mA Useful technical details: Coil No. of Turn Wire Dimension Maximum Current Inductance (mm) (Amp.) (approx.) 800

0.404

0.363

9.2 mH

Theory: For more basic and detailed theory : http://tiny.cc/melde STANDING WAVES IN STRINGS AND NORMAL MODES OF VIBRATION : When a string under tension is set into vibrations, transverse harmonic waves propagate along its length. When the length of string is fixed, reflected waves will also exist. The incident and reflected waves will superimpose to produce transverse stationary waves in the string. The string will vibrate in such a way that the clamped points of the string are nodes and the point of plucking is the antinodes. Let a harmonic wave be set up on a string of length L, fixed at the two ends x=0 and x=L. This wave gets reflected from the two fixed ends of the string continuously and as a result of superimposition of these waves, standing waves are formed on the string. Let the wave pulse moving on the string from left to right be represented by y1 = r sin 2π/ λ (vt - x) Where the symbols have their usual meanings. Note that, here x is the distance from the origin in the direction of the wave (from left to right).It is often convenient to take the origin(x=0) at the interface (the site of reflection), on the right fixed end of the string. In that case, sign of x is reversed because it is measured from the interface in a direction opposite to the incident wave. The equation of incident wave may, therefore, be written as y1 = r sin 2π/ λ (vt + x)

............ (1)

As there is a phase change of π radian on reflection at the fixed end of the string, therefore, the reflected wave pulse travelling from right to left on the string is represented by y2 = r sin [2π/ λ (vt - x) + π ] = - r sin 2π/ λ(vt - x)

............ (2)

According to superposition principle, the resultant displacement y at time t and position x is given by

y = y1 + y2 = r sin 2π/ λ (vt + x) - r sin 2π/ λ (vt - x) = r [sin 2π/ λ(vt + x) - sin 2π/ λ (vt - x)]

….......(3)

Using the relation, sin C - sin D = 2 cos C + D/2 sin C – D/2 We get, y = 2 r cos 2 π v t/ λ sin 2 π x/ λ

……… (4)

As the arguments of trigonometrically functions involved in (4) do not have the form (vt + x), therefore, it does not represent a moving harmonic wave. Rather, it represents a new kind of waves called standing or stationarywaves. At one end of the string, where x = 0 From (4), y = 2 r cos 2 π vt / λ sin 2 π (0)/ λ = 0 At other end of the string, where x = L From (4), y = 2 r cos 2 π vt / λsin 2 π L / λ

.......... (5)

As the other end of the string is fixed, y = 0, at this end For this, from (5), sin 2 π L/ λ = 0 = sin n π, where n = 1,2,3.......... sin 2 π L/ λ = n π λ = 2 L/N

.............(6)

where n = 1, 2, 3..... correspond to 1st, 2nd, 3rd..... normal modes of vibration of the string. (i) First normal mode of vibration : Suppose λ1 is the wavelength of standing waves set up on the string corresponding to n = 1. From (6),

λ1 = 2 L/1

or

L = λ1/2

The frequency of vibration is given by

1  v / 1  v / 2l As v = √T/m where T is the tension in the string and m is the mass per unit length of the string.

1  1 / 2l  T / m 

The string vibrates as a whole in one segment, as shown in figure.

This normal mode of vibration is called fundamental mode. The frequency of vibration of string in this mode is minimum and is called fundamental frequency. The sound or note so produced is called fundamental note or first harmonic. Formula Used: A string can be set into vibrations by means of an electrically maintained tuning fork, thereby producing stationary waves due to reflection of waves at the pulley. The end of the pulley where it touches the pulley and the position where it is fixed to the prong of tuning fork. (i)For the transverse arrangement, the frequency is given by n ฀ 1/ 2l ฀ T / m where ŒL€ is the length of thread in fundamental modes of vibrations, Œ T € is the tension applied to the thread and Œm€ is the mass per unit

length of thread. If Œp€ loops are formed in the length ŒL€ of the thread, then n   p / 2l 

T

/ m

(ii)For For the longitudinal arrangement, when Œp€ loops are formed, the frequency is given by n   p / l

T

/m 

Procedure: 1. Assemble the setup as shown in figure below.

Figure 2: Transverse Arrangement

2. Firstly tide the string (thread), as its one is fixed and other end passes over pulley which is fitted on pulley stand and carries a pan of weights. 3. The core (soft iron rod) of electromagnet should be lies at the center of coil. 4. Now connect mains cord between mains and Melde€s Electrical Vibrator. 5. Take two patch cords from the accessory box and connect the 6 V AC supply to the coil. 6. Now place the permanent magnet with opposite polarities on either side of core as shown in figure. 7. Switch on the AC supply. 8. Now adjust the distance between magnets for getting some vibrations in string. 9. Vibrations of maximum amplitude are obtained by adding some amount of weight in the pan. 10. Note the number of loops p formed in the length L of thread. 11. Length of string can be adjusted with the help of pulley stand for better result. 12. Take the readings for different number of loops by increasing the weight in the pan for fix distance of string. Note: Increase the weight in the pan till the loops are seen clearly. 13. Note the all values in below observation table:

Formula Used:

n

p 2L

T m

Where, p = Number of loops L = Length of thread in fundamental mode of vibrations T = Tension of the string m = Mass per unit length Observations:

Total Length of the thread = ...................cm =........................m Mass of the thread =................................gm =...................... Kg Mass per unit length of the thread (m) =......................Kg/meter (0.0000605 Kg/m approx.) Mass of pan = ...............gm =................. Kg

Observation Table: Tension T Applied S.No.

Total weight = Tension in weight placed string = {Total in pan + weight weight x g} (N) of pan (Kg)

1.

No. of loops Corresponding Frequency Mean frequency of (pi) AC Mains length of the of AC mains thread L meter

ni 

pi T 2L m

n

n1 n2 ...nr r

n1=….. n2=….. n3=….. n4=…..

p1=…. p2=…. p3=…. p4=….

2.

3.

Mean n = .......................Hz

For Longitudinal arrangement: In this case the vibarator is adjusted such that the motion of the rod is in same direction as the length of the thread. The procedure remains the same as described in case of transverse arrangement.

n

p L

T m

Result:

The frequency of A.C. mains, using Transverse arrangement = ...........................Hz Longitudinal arrangement = ........................Hz

Note: To get better result, determine the value of mass per unit length (m) of the experimental thread.

PRECAUTIONS : • The thread should be uniform and inextensible. • Well defined loops should be obtained by adjusting the tension with milligram weights. • Frictions in the pulley should be least possible....